3 Answers
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The battle of Zama was a seesaw fight for much of the battle. The result of a Roman victory resulted largely from the "fortunes of war."

The Carthaginians had more infantry, the Romans more cavalry, but the Carthaginians hoped to turn their elephants to their advantage. This didn't work, because the Roman Scipio, suspecting that the elephants could only charge in a straight line, ordered his men to get out of their way. They couldn't be used in a flanking maneuver that might have hurt the Romans.

The Carthaginian cavalry led the Romans away on a chase-about, but the more numerous Carthaginian infantry got somewhat the better of the Roman infantry. At just the time they were about to make this advantage felt, the Roman cavalry returned and took the Carthaginian infantry in the rear, completing the Roman victory. This happened exactly at the "psychological moment," shortly after the Carthaginian third line had joined the infantry attack. Earlier, they had been held in reserve, and could have been "turned" to face the Roman cavalry.

Essentially, it came down to a race between the Carthaginian infantry versus the Roman cavalry. The elephants were never really a factor. It could have gone the other way, but didn't.

Hannibal wanted to win the battle with his infantry, which was superior to the Romans, and distract the Roman cavalry. Therefore, he wanted the cavalry clash to happen first, the idea being that his cavalry would be routed, the Roman cavalry would pursue them and be out of the battle, and he could start the infantry battle.

Delbrueck here suggests that Hannibal used his elephants as a way of delaying the infantry battle. Honestly, I can't think of another reason for doing it: elephants were generally known not to be all that effective against infantry, and Hannibal already had the advantage there. Elephants were effective against cavalry, and so the normal use would be to disrupt the Roman cavalry to give the Carthaginian cavalry a chance.

Anyway, the elephants were largely ineffective against the Roman infantry, the Roman cavalry pursued the routed Carthaginian cavalry, and Hannibal's infantry was winning the battle. Then the Roman cavalry, instead of pursuing the Carthaginians further, returned and fell on Hannibal's rear, deciding the battle.